Geometric structures and invariants of links and 3-manifolds

链接和 3 流形的几何结构和不变量

• 批准号: 1404754
• 负责人: Efstratia Kalfagianni
• 金额: $ 22.44万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404754&HistoricalAwards=false
• 关键词: Geometric; structures; invariants; links; manifolds

The research of this project falls in the area of 3-dimensional topology. The central objects of study in this area are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3- dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. The solution of Thurston's Geometrization Conjecture has established that 3-manifolds (and complements of knots in them) decompose into pieces that admit explicit geometries and that hyperbolic geometry is the one that appears more often. In practice, however, 3-manifolds are often given in terms of combinatorial topological descriptions and it is both natural and important to seek ways to deduce geometric information from these descriptions. One of the ways that topologists have been approaching the study of 3- manifolds is through the use of invariants. In the last few decades ideas originated in physics led mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. Understanding the connections of topological and combinatorial quantities and invariants to geometry is a central and important goal of low dimensional topology. The main theme of this project is to establish such connections and explore their ramifications and applications to other areas of mathematics.This project will establish relationships between geometry and combinatorial descriptions, properties, and quantum invariants of links and 3-manifolds. The PI has developed a setting for establishing new unexpected relations between the colored Jones link polynomials, the topology and geometry of essential surfaces in link complements, and hyperbolic geometry. One part of the project will continue developing this theory and exploring its applications. Another part, will combine several new techniques, to develop methods for recognizing geometric structures on 3-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data. A third part will study skein link theory in 3-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds. A fourth part will explore the applicability of quantum knot invariants to classical questions in knot theory and search for a classification of crossing changes that do not alter the topology of the underlying knots. The project also involves the research of graduate students currently working with PI.

本项目的研究属于三维拓扑学领域的研究福尔斯。这一领域的中心研究对象是称为三维流形的空间。三维流形是一个局部看起来像普通三维空间的对象，但其整体结构可能是复杂的。三维拓扑学的一个重要部分也是研究结（以某种纠缠的方式嵌入三维流形中的环）及其分类。瑟斯顿的几何化猜想的解决方案已经建立了3-流形（和其中的结的补）分解成允许显式几何的部分，而双曲几何是更经常出现的几何。然而，在实践中，3-流形往往给出的组合拓扑描述，这是自然的，重要的是寻求方法来推导几何信息，从这些描述。拓扑学家研究三维流形的方法之一是使用不变量。在过去的几十年里，源于物理学的思想使数学家们发现了各种各样的纽结和三维流形的不变量。了解拓扑和组合的数量和几何不变量的连接是低维拓扑学的一个中心和重要目标。本项目的主题是建立这样的联系，并探索它们的分支和应用到其他数学领域。本项目将建立几何和组合描述，属性之间的关系，量子不变量的链接和3-流形。PI已经开发了一种设置，用于建立有色琼斯链接多项式之间的新的意想不到的关系，拓扑结构和几何的基本表面在链接补充，和双曲几何。该项目的一部分将继续发展这一理论并探索其应用。另一部分，将结合联合收割机几种新的技术，开发从纯组合输入识别三维流形上几何结构的方法，并从拓扑数据中获得几何量的估计。第三部分将研究3流形中的绞链理论，它的不变量，以及它与3流形的几何分解的相互作用。 第四部分将探讨量子结不变量在结理论中的经典问题的适用性，并寻找一种不改变底层结拓扑结构的交叉变化分类。该项目还涉及目前与PI合作的研究生的研究。